## Abstract

Proper use of masking in subtractive color photography is dependent upon a means of establishing the color reproduction equations which give the values of the mask gammas. Yule has shown that such equations can be obtained from an extension of the principles of duplicating, while MacAdam has done so by treating the subtractive system in terms of an equivalent additive system. Marriage has demonstrated that a specification of the requirements necessary for the exact reproduction of any four selected colors can be used to derive a unique set of equations. In the present paper, it is shown that Marriage’s method can be extended to include colors in any number greater than four, provided “approximate reproduction” rather than “exact reproduction” is taken as the criterion. The resulting equations will depend upon the manner in which the criterion of approximate reproduction is applied, as well as upon the particular selection of colors to be reproduced. While these aspects of the problem have not been investigated, results have been obtained which appear to be reasonable. The method may be applied to any assumed types of sensitivity distributions and forms of color reproduction equations.

© 1949 Optical Society of America

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### Equations (13)

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(1)
$$\begin{array}{ll}\hfill c& ={\gamma}_{cr}(-logR)+{\gamma}_{cg}(-logG)+{\gamma}_{cb}(-logB)+{K}_{c};\hfill \\ \hfill m& ={\gamma}_{mr}(-logR)+{\gamma}_{mg}(-logG){\gamma}_{mb}(-logB)+{K}_{m};\hfill \\ \hfill y& ={\gamma}_{yr}(-logR)+{\gamma}_{yg}(-logG){\gamma}_{yb}(-logB)+{K}_{y};\hfill \end{array}$$
(2)
$$\begin{array}{lll}R=\mathit{\int}{S}_{r}\mathit{\text{THd}}\mathrm{\lambda},\hfill & G=\mathit{\int}{S}_{g}\mathit{\text{THd}}\mathrm{\lambda},\hfill & B=\mathit{\int}{S}_{b}\mathit{\text{THd}}\mathrm{\lambda},\hfill \end{array}$$
(3)
$$\begin{array}{lll}{D}_{r}=-logR,\hfill & {D}_{g}=-logG,\hfill & {D}_{b}=-logB,\hfill \end{array}$$
(4)
$$\mathit{\int}{S}_{r}Hd\mathrm{\lambda}=\mathit{\int}{S}_{g}Hd\mathrm{\lambda}=\mathit{\int}{S}_{b}Hd\mathrm{\lambda}=1.$$
(5)
$$\begin{array}{ll}\hfill c& ={\gamma}_{cr}{D}_{r}+{\gamma}_{cg}{D}_{g}+{\gamma}_{cb}{D}_{b}+{K}_{c},\hfill \\ \hfill m& ={\gamma}_{mr}{D}_{r}+{\gamma}_{mg}{D}_{g}+{\gamma}_{mb}{D}_{b}+{K}_{m},\hfill \\ \hfill y& ={\gamma}_{yr}{D}_{r}+{\gamma}_{yg}{D}_{g}+{\gamma}_{yb}{D}_{b}+{K}_{y}.\hfill \end{array}$$
(6)
$$\begin{array}{ll}\hfill c& =0.24{D}_{r}-1.66{D}_{g}+2.42{D}_{b},\hfill \\ \hfill m& =-0.23{D}_{r}+2.27{D}_{g}-1.04{D}_{b},\hfill \\ \hfill y& =0.52{D}_{r}+0.18{D}_{g}+0.30{D}_{b}.\hfill \end{array}$$
(7)
$$\begin{array}{ll}\hfill c& =1.866{D}_{r}-0.783{D}_{g}-0.070{D}_{b}+0.121,\hfill \\ \hfill m& =-0.649{D}_{r}+2.151{D}_{g}-0.487{D}_{b}-0.019,\hfill \\ \hfill y& =-0.530{D}_{r}-0.362{D}_{g}+1.895{D}_{b}-0.098.\hfill \end{array}$$
(8)
$$\begin{array}{ll}\hfill c& =1.013{D}_{n}+0.121,\hfill \\ \hfill m& =1.015{D}_{n}-0.019,\hfill \\ \hfill y& =1.003{D}_{n}-0.098.\hfill \end{array}$$
(9)
$$\begin{array}{ll}\hfill {\gamma}_{cr}+{\gamma}_{cg}+{\gamma}_{cb}& =1,\hfill \\ \hfill {\gamma}_{mr}+{\gamma}_{mg}+{\gamma}_{mb}& =1,\hfill \\ \hfill {\gamma}_{yr}+{\gamma}_{yg}+{\gamma}_{yb}& =1.\hfill \end{array}$$
(10)
$$\begin{array}{ll}\hfill c& =1.846{D}_{r}-0.894{D}_{g}+0.048{D}_{b},\hfill \\ \hfill m& =-0.651{D}_{r}+2.154{D}_{g}-0.503{D}_{b},\hfill \\ \hfill y& =-0.516{D}_{r}-0.287{D}_{g}+1.803{D}_{b}.\hfill \end{array}$$
(11)
$$\begin{array}{ll}\hfill c& ={\gamma}_{cr}{D}_{r}+{K}_{c},\hfill \\ \hfill m& ={\gamma}_{mg}{D}_{g}+{K}_{m},\hfill \\ \hfill y& ={\gamma}_{yb}{D}_{b}+{K}_{y},\hfill \end{array}$$
(12)
$$\begin{array}{ll}\hfill c& ={\gamma}_{cr}{D}_{r},\hfill \\ \hfill m& ={\gamma}_{mr}{D}_{r}+{\gamma}_{mg}{D}_{g},\hfill \\ \hfill y& ={\gamma}_{yg}{D}_{g}+{\gamma}_{yb}{D}_{b}.\hfill \end{array}$$
(13)
$$\begin{array}{ll}\hfill c& ={D}_{r},\hfill \\ \hfill m& =-0.65{D}_{r}+1.65{D}_{g},\hfill \\ \hfill y& =-0.80{D}_{g}+1.80{D}_{b}.\hfill \end{array}$$